Gauss's Law for magnetism is ∇ ⋅ B = 0This allows us to write the magentic field B as the curl of another field the magnetic vector potential, A. B = ∇ × AThis adhers to ∇ ⋅ ( ∇ × A ) = 0However, if a monopole does exist then we have ∇ ⋅ B = ρ m Where ρ m is some magnetic charge density however with a magentic vector potential this violates the equation, ∇ ⋅ ( ∇ × A ) ≠ 0Does that mean if magnetic monopoles does exist, that the magnetic field can no longer be defined by a magnetic vector potential? In which case how was dirac able to still define the magnetic field by a magnetic vector potentials?

Question

Gauss&#039;s Law for magnetism is  ∇  ⋅  B  =  0This allows us to write the magentic field B as the curl of another field the magnetic vector potential, A.  B  =  ∇  ×  AThis adhers to   ∇  ⋅  (  ∇  ×  A  )  =  0However, if a monopole does exist then we have  ∇  ⋅  B  =      ρ          m      Where       ρ          m       is some magnetic charge density however with a magentic vector potential this violates the equation,   ∇  ⋅  (  ∇  ×  A  )  ≠  0Does that mean if magnetic monopoles does exist, that the magnetic field can no longer be defined by a magnetic vector potential? In which case how was dirac able to still define the magnetic field by a magnetic vector potentials?

Lea Johnson · Accepted Answer

That means that, in general, you will have to write       B    =  ∇  ψ  +  ∇  ×      A  . Then   ∇  ⋅      B    =      ∇    2    ψ  =      ρ    m  . Of course   ∇  ⋅  (  ∇  ×      A    )  =  0 for any vector field A, as       d    2    =  0

Gauss's Law for magnetism is <mi mathvariant="normal">∇ ⋅

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